By the classical result of Erdős and Szekeres [1]} concerning monotone subsequences, we get \(t^{(r)}(n)= \Omega (\sqrt{n})\) . Indeed, in a monotone permutation any \(r\)  disjoint subsequences of the same length yield \(r\) -twins. For \(r=2\) , using a probabilistic argument, Gawron [2]} proved that \(t^{(2)}(n)=O(n^{2/3})\) . He conjectured that this bound is tight, namely, that we also have \(t^{(2)}(n)=\Omega (n^{2/3})\) . A currently best result towards this conjecture was obtained recently by Bukh and Rudenko [3]} (see also [4]}) who showed that \(t^{(2)}(n)=\Omega (n^{3/5})\) .
